#8709284

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#40723	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4255 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,10)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(33,34,38,42)(35,39,40,36)(37,41,47,48)(43,45,46,44);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,18,22,23)(19,21,25,30)(20,24,28,31)(26,29,32,27)(33,35)(34,39)(36,42)(37,43)(38,40)(41,45)(44,48)(46,47);(4,13)(8,12)(11,14)(15,16)(18,23)(19,25)(24,31)(26,32)(33,36,38,39)(34,40,42,35)(37,44,47,45)(41,46,48,43);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,19)(18,21)(20,26)(22,25)(23,30)(24,29)(27,31)(28,32)(33,36)(34,35)(37,45)(38,39)(40,42)(41,46)(43,48)(44,47);(1,5,6,3)(2,7,9,10)(4,14,13,11)(8,15,12,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,35)(34,36)(37,46)(38,40)(39,42)(41,45)(43,47)(44,48);(1,2)(3,7)(4,12)(5,10)(6,9)(8,13)(11,16)(14,15)(17,21)(18,25)(19,23)(20,27)(22,30)(24,26)(28,29)(31,32)(33,37)(34,41)(35,43)(36,44)(38,47)(39,45)(40,46)(42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8616480);